Are there moments in a child's development when he is particularly ready to make the transition from one knowledge state to another? Previous work by the investigator has shown that the mismatch (discordance) between spontaneous gesture and the speech it accompanies can be used to determine whether a child is in an unstable knowledge state. Moreover, children whose knowledge of a concept is assessed, via discordance, as unstable have been found to be particularly likely to benefit from instruction in that concept, and thus appear to be on the threshold of a transition to a new knowledge state. Our aim is to extend our understanding of discordance in 3 ways: (1) to determine whether discordance is an index of the instability of a child's knowledge of a particular concept and not a characteristic of the child himself; (2) to determine whether the discordant (unstable) state is transitional in the sense that it is both preceded and followed by a concordant (stable) state; (3) to determine whether a child in a discordant state possesses implicit information that he has not yet integrated with his other knowledge; this implicit knowledge will be tapped in two ways, by analyzing the information conveyed in a child's discordant responses, and by probing his proximal zone. To meet the first of these goals, 30 children (ages 10-11) will be tested on 2 different concepts, conservation and mathematical equivalence, and the discordance status of each child will be assessed with respect to both concepts. To achieve the second and third goals, the relationship between changing knowledge and discordance will be explored in these same 2 concepts using 2 data sets: (1) data already collected on 44 children, ages 5-8, and 80 children, ages 9-12, and (2) longitudinal data to be collected on 46 children, ages 5-7, and 23 children, ages 9-11. The populations for each study will be drawn from a variety of ethnic and socio-economic backgrounds. Each study will be videotaped. The objective of these studies is to gain a more complete understanding of discordance as an index of transitional knowledge, and thus to contribute toward a general understanding of knowledge states, both static and in flux.